Problem 22 - Calculus with Concepts in Calculus

Question

Solve the inequality. $$ -0.1<3 x+4<0.1 $$

Step-by-Step Solution

Answer

The solution is \(-\frac{4.1}{3} < x < -\frac{3.9}{3}\).

Step 1: Split the Inequality

The given double inequality \[-0.1 < 3x + 4 < 0.1\] can be split into two separate inequalities:1) \(-0.1 < 3x + 4\)2) \(3x + 4 < 0.1\).These can be solved individually and the solution sets must satisfy both conditions.

Step 2: Solve the First Inequality

Start with the first inequality:\[-0.1 < 3x + 4\]Subtract 4 from both sides:\[-0.1 - 4 < 3x\]Simplify:\[-4.1 < 3x\]Now, divide both sides by 3:\[-\frac{4.1}{3} < x\]Thus, we have:\[x > -\frac{4.1}{3}\].

Step 3: Solve the Second Inequality

Now solve the second inequality:\[3x + 4 < 0.1\]Subtract 4 from both sides:\[3x < 0.1 - 4\]Simplify:\[3x < -3.9\]Divide both sides by 3:\[x < -\frac{3.9}{3}\]Thus, we have:\[x < -\frac{3.9}{3}\].

Step 4: Combine the Solutions

The solutions to the two inequalities are:1) \(x > -\frac{4.1}{3}\)2) \(x < -\frac{3.9}{3}\)For both conditions to be true simultaneously, the solution set is the intersection \[-\frac{4.1}{3} < x < -\frac{3.9}{3}\].

Key Concepts

Double Inequalities

A double inequality is a mathematical expression involving two inequality signs, encapsulating a single expression or variable in between. In the inequality \[-0.1 < 3x + 4 < 0.1\], you have one expression, \(3x + 4\), that must simultaneously satisfy two conditions. This might seem complex at first, but a helpful strategy is to break it into two separate inequalities, making them more manageable:

• \(-0.1 < 3x + 4\)
• \(3x + 4 < 0.1\).

By solving these independently and considering their combined solutions, you gain a clear understanding of the problem.
Through this process, double inequalities provide a concise method to handle problems involving experimental limits or specific parameter ranges in real-life applications.

Inequality Solutions

Solving inequalities involves finding all values of a variable that make the inequality true. For the broken down inequalities from our example:

• \(-0.1 < 3x + 4\)
• \(3x + 4 < 0.1\)

Each can be solved following a step-by-step algebraic process:

1. **For \(-0.1 < 3x + 4\):** - Subtract 4 from both sides to isolate the term with \(x\): \(-0.1 - 4 < 3x\). - Simplify it: \(-4.1 < 3x\). - Divide both sides by 3 to solve for \(x\): \(-\frac{4.1}{3} < x\). - This tells us that \(x\) must be greater than \(-\frac{4.1}{3}\).2. **For \(3x + 4 < 0.1\):** - Again, subtract 4 to isolate \(3x\): \(3x < 0.1 - 4\). - Simplify it: \(3x < -3.9\). - Divide by 3: \(x < -\frac{3.9}{3}\). - This means \(x\) must be less than \(-\frac{3.9}{3}\).
Understanding these steps is fundamental to mastering inequalities as it shows not only how the inequality signs work but also how to manipulate algebraic expressions to find a solution set.

Intersection of Solution Sets

The intersection of solution sets is where these individual solutions overlap, meeting all the required conditions of the original double inequality. For the inequality \(-0.1 < 3x + 4 < 0.1\), you split it into:

• \(x > -\frac{4.1}{3}\)
• \(x < -\frac{3.9}{3}\)

Once you derive these conditions, finding their intersection involves identifying which values of \(x\) satisfy both inequalities simultaneously.
For example, both solutions must hold true, meaning \(x\) is strictly greater than \(-\frac{4.1}{3}\) and less than \(-\frac{3.9}{3}\).
These intervals don't overlap, suggesting there is no real number satisfying both conditions at the same time. However, understanding the union and intersection of sets helps immensely in scenarios involving ranges and bounds.
Thinking of the intersection as needing to meet *both* conditions simultaneously ensures thoroughness in finding valid solutions.